Multi-Message Private Information Retrieval: Capacity Results and Near-Optimal Schemes
This work addresses privacy in data retrieval for users accessing multiple messages from databases, with incremental improvements in capacity bounds.
The paper tackles the problem of multi-message private information retrieval (MPIR) from replicated databases, determining exact sum capacity for cases where the number of desired messages is at least half the total, and providing tight bounds for other cases with differences as small as 0.0082, showing joint retrieval is more efficient than single-message schemes.
We consider the problem of multi-message private information retrieval (MPIR) from $N$ non-communicating replicated databases. In MPIR, the user is interested in retrieving $P$ messages out of $M$ stored messages without leaking the identity of the retrieved messages. The information-theoretic sum capacity of MPIR $C_s^P$ is the maximum number of desired message symbols that can be retrieved privately per downloaded symbol. For the case $P \geq \frac{M}{2}$, we determine the exact sum capacity of MPIR as $C_s^P=\frac{1}{1+\frac{M-P}{PN}}$. The achievable scheme in this case is based on downloading MDS-coded mixtures of all messages. For $P \leq \frac{M}{2}$, we develop lower and upper bounds for all $M,P,N$. These bounds match if the total number of messages $M$ is an integer multiple of the number of desired messages $P$, i.e., $\frac{M}{P} \in \mathbb{N}$. In this case, $C_s^P=\frac{1-\frac{1}{N}}{1-(\frac{1}{N})^{M/P}}$. The achievable scheme in this case generalizes the single-message capacity achieving scheme to have unbalanced number of stages per round of download. For all the remaining cases, the difference between the lower and upper bound is at most $0.0082$, which occurs for $M=5$, $P=2$, $N=2$. Our results indicate that joint retrieval of desired messages is more efficient than successive use of single-message retrieval schemes.